Related papers: Leading coefficient in the Hankel determinants rel…
The Schinzel hypothesis essentially claims that finitely many irreducible polynomials in one variable over Z simultaneously assume infinitely many prime values unless there is an obvious reason why this is impossible. We prove that under a…
As a particular one parameter deformation of the quantum determinant, we introduce a quantum $\alpha$-determinant and study the $\mathcal{U}_q(\mathfrak{gl}_n)$-cyclic module generated by it: We show that the multiplicity of each…
The Hamburger moment problem for the $q$-Lommel polynomials which are related to the Hahn-Exton $q$-Bessel function is known to be indeterminate for a certain range of parameters. In this paper, the Nevanlinna parametrization for the…
In this paper we consider the weighted q-Bernoulli numbers and polynomials which are differnt type of Carlitz's q-Bernoulli numbers and polynomials. From these numbers and polynomials, we derive some interesting formulaes and identities.
In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is…
The notion of generalized quantum monoids is introduced. It is proved that the quantum coordinate ring of the monoid can be lifted to a quantum hyper-algebra, in which the quantum determinant and quantum Pfaffian are sent to the quantum…
In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Ap\'ery numbers. For any nonnegative integer $n$, define…
Gaussian binomial coefficients are q-analogues of the binomial coefficients of integers. On the other hand, binomial coefficients have been extended to finite words, i.e., elements of the finitely generated free monoids. In this paper we…
In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class $\mathcal{S}$ of univalent functions in the unit disc.
We study $n\times n$ Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We…
Given the rational power series $h(x) = \sum_{i \geq 0} h_i x^i \in \mathbb{C}[[x]]$, the Hankel determinant of order $n$ is defined as $H_n(h(x)) := \det (h_{i+j})_{0 \leq i,j \leq n-1}$. We explore the relationship between the Hankel…
The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of…
Given a polynomial P of partial derivatives of the Kahler metric, expressed as a linear combination of directed multigraphs, we prove a simple criterion in terms of the coefficients for $P$ to be an invariant polynomial, i.e. invariant…
We develop a simple method for constructing polynomial invariants of degree 4 for even-$n$ qubits and give explicit expressions for these polynomial invariants. We demonstrate the invariance of the polynomials under stochastic local…
Lascoux polynomials have been recently introduced to prove polynomiality of the maximum-likelihood degree of linear concentration models. We find the leading coefficient of the Lascoux polynomials (type C) and their generalizations to the…
In this paper, we found new q-binomial formula for Q-commutative operators. Expansion coefficients in this formula are given by q-binomial coefficients with two bases (q,Q), determined by Q-commutative q-Pascal triangle. Our formula…
Polynomial relations between the generators of $q$--deformed Heisenberg algebra invariant under the quantization and $q$-deformation are discovered. One of the examples of such relations is the following: if two elements $a$ and $b$,…
We develop a coordinate-free probabilistic framework for determinantal point processes associated with Bergman kernels on compact complex manifolds. The basic issue is that Bergman kernels are naturally line-bundle-valued:…
We present an asymmetric $q$-Painlev\'e equation. We will derive this using $q$-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this $q$-Painlev\'e equation (up to a simple…
We derive a general expression for the Hankel determinants of a Dirichlet series F(s) and derive the asymptotic behavior for the special case that F(s) is the Riemann zeta function. In this case the Hankel determinant is a discrete analogue…